Quadratics are the point where arithmetic turns into algebra. A question rarely announces "solve a quadratic" - it hides one inside a story about the area of a field, the product of two consecutive numbers, or a profit curve. The candidate who recognises the squared unknown early has a routine; the candidate who does not will try to guess numbers until time runs out.
The section rewards two skills: turning a paragraph into the standard form a x² + b x + c = 0, and reading the coefficients for what they already say about the roots.
The model omits one case deliberately: not every quadratic factorises over the rational numbers, so the formula is not a lazy fallback - it is the general method, and the discriminant tells you which situation you are in.
To factor, find two numbers whose product is a times c and whose sum is b. When a = 1 this reduces to two numbers multiplying to c and adding to b. For x² − 5x + 6, the pair 2 and 3 multiply to 6 and add to 5 with a negative sign, so the factors are (x − 2)(x − 3) and the roots are 2 and 3.
Substitute once, compute the discriminant b² − 4ac, then treat the plus and minus branches separately.
| Discriminant | Roots | Graph meets x-axis |
|---|---|---|
| D > 0 | Two distinct real roots | Twice |
| D = 0 | One repeated real root | Once, at the vertex |
| D < 0 | No real roots | Never |
Forming an equation from its roots is the reverse: if the roots are p and q, then x² − (p + q)x + pq = 0.
Solve 2x² + 3x − 2 = 0.
Here a = 2, b = 3 and c = −2. Compute the discriminant first:
D > 0, so there are two distinct real roots. Since √25 = 5, the formula gives:
The plus branch gives x = 2 ÷ 4 = 0.5. The minus branch gives x = −8 ÷ 4 = −2.
Check against the coefficient identities. The roots sum to 0.5 + (−2) = −1.5, and −b ÷ a = −3 ÷ 2 = −1.5. Their product is 0.5 × (−2) = −1, and c ÷ a = −2 ÷ 2 = −1. Both match, so the roots are correct.
Factorising confirms it: 2x² + 3x − 2 = (2x − 1)(x + 2), which is zero at x = 0.5 and at x = −2.
What is the discriminant of x² + 4x + 4 = 0?
For 3x² − 6x + 2 = 0, what are the sum and product of the roots?
What are the roots of x² − 7x + 12 = 0?
Which equation has roots 5 and −2?
Without looking back: what does the discriminant tell you, and how do the sum and product of the roots relate to a, b and c?
The sign of D decides the number of real roots; the roots sum to −b ÷ a and multiply to c ÷ a.
For x² − 5x + 6 = 0 the roots are 2 and 3, so their sum is and their product is . The discriminant of x² + 4x + 4 is , so the number of distinct real roots is .
A rectangular garden is 3 metres longer than it is wide, and its area is 40 square metres. Find its dimensions, and say what a negative root would mean here.
Let the width be w metres, so the length is w + 3. Area gives w(w + 3) = 40, which rearranges to w² + 3w − 40 = 0.
Factorise: the pair 8 and −5 multiplies to −40 and adds to 3, so (w + 8)(w − 5) = 0 and w = 5 or w = −8.
A width cannot be negative, so w = 5 and the length is 8. Check the area: 5 × 8 = 40 square metres.
The negative root is not an error; it is the algebra reporting a second algebraic solution that the physical story rules out. Always re-read the last line of the question to see whether a negative (or fractional) root is admissible.
Quadratics describe one unknown squared against the same unknown. Sequences and series describe an ordered list of quantities whose pattern can be added up. The next lesson gives you the two patterns that carry almost all of those questions - arithmetic and geometric.